You can essentially spend an entire lifetime trying different bullet, powder, ect combinations with the '06 and about the time you think you've got is "worked out", something new will come along.
However, when you brought your .243 into the equation, I can only think of 3 powders that will maximize both......
first, try IMR4350. My .243 didn't like it however.
Then, IMR4831, my .243 was "so-so" with this w/100gr bullets, not so hot with lighter bullets.
But, Reloader 22 "hit the spot", ususally near 1-hole accuracy with either 100gr Hornady or Noslers at 42.5gr (Noslers max and best accuracy load).
I found my '06's, that 60.0gr of RL-22 was best, and with my brass, I can't get 61.0 in the case and seat a Nosler B.T. or Hornady SST without damaging the bullets. However, 60.0gr gives me 2,850fps w/180gr bullet which is .300H&H mag. levels. Kicks like it too !!! (24"bbls).
For Whitetails, I prefer a 150gr bullet in the '06, and either IMR4064 or Reloader15 have given best accuracy. If a Sierra or Nosler 150gr (doesn't matter which one's !!) and 52.0gr of 4064 or RL15 for 2,950-3,000fps don't give MOA, your gun's not capable. I've also gotten 1-hole accuracy with 165gr bullets and IMR4350 (56.5-57.5gr). But, inside 300yds, the 150's shoot flatter. The only .30cal I use a 180gr in is a .300RUM, and because it takes this much bullet weight to make it "behave" in the accuracy realm. Also, energy and trajectory are superior in the +300yd realm where this gun is used. BTW, I wouldn't use this for anything I wouldn't use the '06 for.
FWIW; my brother uses a pre-64 Win. mod70 FW in .308 with 150gr Nosler Partitions for "everything". He's taken elk, mule deer, whitetail, pronghorn, and sheep with this load and has yet to recover a bullet, even from an angled body shot on a 900lb elk. He prefers H4895 and gets ~2,900fps. His longest shot has been a whitetail in Georgia, too!!!!! about 275yds, so, he no longer sights in at 300yds, just 250..... and uses a dead-on hold to ~300yds.
If you need more gun than an '06, you need at least a .375H&H, and perhaps even more............BTDT.
You'll probably find out as I did as early as the early '80's that the '06 is a bit "much" for whitetails and go back to your .243. I personnally prefer a .257Roberts or .30/30 for deer. Even my .35Rem is a bit "hard" on them.